A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

1, 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 112, 210, 387, 728, 1360, 2570, 4845, 9198, 17459, 33288, 63519, 121574, 232960, 447392, 860265, 1657009, 3195465, 6170930, 11930100, 23091222, 44738560, 86767016, 168428805, 327235602, 636289024, 1238188770, 2411205111, 4698767640, 9162588158, 17878237850

Offset

1,5

Comments

Arndt's PARI code computes a(n) as the sum, divided by n, of every 3rd term in row n of L = A050186 = Möbius transform of binomials, starting with k = (1-n) mod 3 (nonnegative remainder), where k = 0 and k = n give L(n, k) = 0 and can be omitted. Cf. A053727, EXAMPLE and second PROGRAM. - M. F. Hasler, Sep 27 2018

Links

Robert Israel, Table of n, a(n) for n = 1..3331

J. E. Iglesias, Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.

J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.

Formula

a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).

Example

Illustrating computation via L = A050186, cf. COMMENTS: a(1) = [L(1,0)] = 0. a(2) = [L(2,2)] = 0. a(3) = L(3,1)/3 = 3/3 = 1. a(4) = ([L(4,0)] + L(4,3))/4 = 4/4 = 1. a(5) = (L(5,2) + [L(5,5)])/5 = 10/5 = 2. In [...] are terms L(n,0) = L(n,n) = 0.

Maple

f:= proc(n) local D, d;
  D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
  add(numtheory:-mobius(n/d)*(2^d - (-1)^d), d=D)/(3*n)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 14 2019

Mathematica

a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

Prog

(PARI)
L(n, k) = sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
vector(55, n, a(n))
/* Joerg Arndt, Jun 28 2012 */
(PARI) A165920(n, k=(1-n)%3)=sum(i=0, (n-k)\3, A050186(n, k+3*i))\n \\ For illustration. - M. F. Hasler, Sep 30 2018

Crossrefs

This sequence is the half of A165912 (the number of alternate polynomials). A001037 is the enumeration by degree of the polynomials of I. A000048 is the number of 3-elements orbits of S3 action on I.

Sequence in context: A093126 A003237 A191519 * A274160 A190501 A026021

Adjacent sequences: A165917 A165918 A165919 * A165921 A165922 A165923

Keyword

easy,nonn

Author

Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009

Status

approved